The Isoperimetric Problem in Regular Trees

We investigate the inner vertex-isoperimetric problem on the $d$-regular tree $T_d$. We first determine the exact value of the inner vertex-isoperimetric profile $I_d(k)=\min \{\mathrm{\mid }\mathrm{\partial }D\mathrm{\mid }\mid D\subset T_d\text{ finite and connected},\mathrm{\mid }D\mathrm{\mid }=k\}$, and we then introduce a boundary invariant, called the boundary branching excess $\tau (D)$, and show that it provides a simple criterion for optimality. A domain $D\subset T_d$ is shown to be isoperimetrically optimal if and only if $\tau (D)\le d-2$. Finally, we show that every domain in $T_d$ admits a canonical decomposition as an iterated gluing of full domains, namely domains whose entire boundary consists of leaves. This yields a complete description of all inner vertex-isoperimetric minimizers in $T_d$.